Inside was dark with just enough light to see shapes, and silence was
playing in the background in an endless loop. The source was a green
neon light attached to the rusty wall in front of the bar. There were
only two persons sitting there; a young man with short hair and a girl
with long black hair and smooth facial features. No one else was
there; not even the bartender. They were glancing at each other from
time to time. Both were holding glasses. He was having a whisky, and
she a martini.

I also would like to improve/or fix bad writing style in this opening dialog:

“I had a nice time,” he said while staring at his glass.
“Me too,” she said while looking at him with a smile.
“I really mean it,” he said while looking into her eyes. “I feel like...for the first time, someone listens to not only to my dreams, but also my nightmares when I talk about them.”
He turned his eyes into the glass again as having said something inappropriate.
She remained silent and her face turned red. She was trying to say something but it didn’t come out.
Silence continued playing in the background.
"Can I talk to you about something that terrifies me?" he said after a while.
She looked straight into his eyes and nodded with her lips half open.

Is there anything I'm doing wrong or is there something I can do to improve them?

Disclaimer: I'm not a native English speaker and I haven't corrected the grammar.

1 Answer
1

I am not a fan of these "denoting something by describing that which does not exist" techniques, e.g.:

"Yes," he said, unsmiling.

The silence was very loud.

etc. While it does say something, I feel it is left too much to the imagination. Certainly it can be used in specific cases, but as a general rule, only to create that specific feeling of uncertainty.

I do not see the melancholy and darkness here. There's awkwardness in their dialogue, but that's all. I would go straight for the people:

"I had a nice time," he said, staring at his glass of whisky. "Me
too," she replied. He glanced up and saw that she was smiling.
"I really mean it." ... etc They were alone in the bar. The ice in
his glass clinked very loudly in the silence. A faded sign above the
counter said "[something crass]", and he found himself staring at it.
Outside on the street he saw a woman tossing the corpse of a cat into
a shopping cart, and then continue down the street, pushing her cart.

Writing something like "There were only two people sitting there" implies that the reader should have knowledge about how many people ought to be sitting there, and I feel that it sort of jars my reading. Also, it is a bit clunky to first mention it is a bar, then mention the bar in a back reference: "there". "They were alone in the bar." does all this in one fell swoop, at the same time feeling like a very natural observation.

Mentioning short hair on a man is usually redundant, because that is more or less the default image that pops into your mind when you read "a young man". Similarly, a girl would have smooth facial features. I would focus on describing that which is unusual about them, and try to do it innocuously:

"Can I talk to you about something that terrifies me?" he said. He
found it hard to look directly at her. Her smooth skin, long black
hair and startlingly blue eyes made something inside him hurt and
burn.